scholarly journals GRAVITATIONAL OBSERVABLES, INTRINSIC COORDINATES, AND CANONICAL MAPS

2009 ◽  
Vol 24 (10) ◽  
pp. 725-732 ◽  
Author(s):  
J. M. PONS ◽  
D. C. SALISBURY ◽  
K. A. SUNDERMEYER
Keyword(s):  
Phase Space ◽  
Coordinate System ◽  
Taylor Expansion ◽  
Gravitational Theory ◽  
Poisson Brackets ◽  
Canonical Transformations ◽  
Geometric Proof ◽  
Gauge Transformations ◽  
Dirac Brackets ◽  
Generally Covariant

It is well known that in a generally covariant gravitational theory the choice of spacetime scalars as coordinates yields phase-space observables (or "invariants"). However, their relation to the symmetry group of diffeomorphism transformations has remained obscure. In a symmetry-inspired approach we construct invariants out of canonically induced active gauge transformations. These invariants may be interpreted as the full set of dynamical variables evaluated in the intrinsic coordinate system. The functional invariants can explicitly be written as a Taylor expansion in the coordinates of any observer, and the coefficients have a physical and geometrical interpretation. Surprisingly, all invariants can be obtained as limits of a family of canonical transformations. This permits a short (again geometric) proof that all invariants, including the lapse and shift, satisfy Poisson brackets that are equal to the invariants of their corresponding Dirac brackets.

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

scholarly journals On the Dirac eigenvalues as observables of the on-shell N = 2D = 4 Euclidean supergravity

Open Physics ◽  
2008 ◽  
Vol 6 (4) ◽  
Author(s):  
Ion Vancea
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Gauge Transformations

AbstractWe generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.

scholarly journals DIRAC BRACKETS FROM MAGNETIC BACKGROUNDS

2007 ◽  
Vol 04 (04) ◽  
pp. 523-532 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Phase Space ◽  
Symplectic Form ◽  
Weak Magnetic Field ◽  
Complex Geometry ◽  
Poisson Brackets ◽  
Generalized Complex ◽  
Dirac Brackets ◽  
Classical Phase Space

In symplectic mechanics, the magnetic term describing the interaction between a charged particle and an external magnetic field has to be introduced by hand. On the contrary, in generalized complex geometry, such magnetic terms in the symplectic form arise naturally by means of B-transformations. Here we prove that, regarding classical phase space as a generalized complex manifold, the transformation law for the symplectic form under the action of a weak magnetic field gives rise to Dirac's prescription for Poisson brackets in the presence of constraints.

The Darboux coordinate system and Holstein–Primakoff representations on Kähler manifolds

2006 ◽  
Vol 84 (10) ◽  
pp. 891-904
Author(s):  
J R Schmidt
Keyword(s):  
Coordinate System ◽  
Lie Groups ◽  
Lie Algebra ◽  
Poisson Structure ◽  
Kähler Manifolds ◽  
Coadjoint Orbits ◽  
Kahler Manifolds ◽  
Canonical Transformations ◽  
Kahler Geometry

The Kahler geometry of minimal coadjoint orbits of classical Lie groups is exploited to construct Darboux coordinates, a symplectic two-form and a Lie–Poisson structure on the dual of the Lie algebra. Canonical transformations cast the generators of the dual into Dyson or Holstein–Primakoff representations.PACS Nos.: 02.20.Sv, 02.30.Ik, 02.40.Tt

scholarly journals CANONICAL TRANSFORMATIONS IN THREE-DIMENSIONAL PHASE-SPACE

2009 ◽  
Vol 24 (25n26) ◽  
pp. 4769-4788 ◽  
Author(s):  
TEKİN DERELİ ◽  
ADNAN TEĞMEN ◽  
TUĞRUL HAKİOĞLU
Keyword(s):  
Phase Space ◽  
Canonical Transformation ◽  
Generating Functions ◽  
General Framework ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Canonical Transformations ◽  
Dimensional Phase Space ◽  
Nambu Bracket ◽  
Definition Of

Canonical transformation in a three-dimensional phase-space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed based on canonoid transformations. It is shown that generating functions, transformed Hamilton functions and the transformation itself for given generating functions can be determined by solving Pfaffian differential equations corresponding to that quantities. Types of the generating functions are introduced and all of them are listed. Infinitesimal canonical transformations are also discussed. Finally, we show that the decomposition of canonical transformations is also possible in three-dimensional phase space as in the usual two-dimensional one.

scholarly journals SYMPLECTIC MATRIX, GAUGE INVARIANCE AND DIRAC BRACKETS FOR SUPER-QED

1999 ◽  
Vol 14 (29) ◽  
pp. 4687-4704
Author(s):  
D. T. ALVES ◽  
E. S. CHEB-TERRAB
Keyword(s):  
Gauge Invariance ◽  
Zero Mode ◽  
Matrix Approach ◽  
Supersymmetric Extension ◽  
Gauge Transformations ◽  
Symplectic Matrix ◽  
Dirac Brackets

The calculation of Dirac brackets (DB) using a symplectic matrix approach but in a Hamiltonian framework is discussed, and the calculation of the DB for the supersymmetric extension of QED (super-QED) is shown. The relation between the zero-mode of the pre-symplectic matrix and the gauge transformations admitted by the model is verified. A general prescription to construct Lagrangians linear in the velocities is also presented.

scholarly journals QUANTUM-DEFORMED CANONICAL TRANSFORMATIONS, W∞ ALGEBRAS AND UNITARY TRANSFORMATIONS

1994 ◽  
Vol 09 (32) ◽  
pp. 5801-5820 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Phase Space ◽  
Pseudodifferential Operators ◽  
Star Product ◽  
Canonical Transformations ◽  
Symbol Calculus ◽  
Algebra Of Functions ◽  
Unitary Transformations ◽  
Symbols Of Operators ◽  
Left And Right ◽  
Gns Construction

We investigate the algebraic properties of the quantum counterpart of the classical canonical transformations using the symbol calculus approach to quantum mechanics. In this framework we construct a set of pseudodifferential operators which act on the symbols of operators, i.e. on functions defined over phase space. They act as operatorial left and right multiplication and form a W∞×W∞ algebra which contracts to its diagonal subalgebra in the classical limit. We also describe the Gel’fand-Naimark-Segal (GNS) construction in this language and show that the GNS representation space (a doubled Hilbert space) is closely related to the algebra of functions over phase space equipped with the star product of the symbol calculus.

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